Application of symmetric spaces and Lie triple systems in numerical analysis

Symmetric spaces are well known in differential geometry from the study of spaces of constant curvature. The tangent space of a symmetric space forms a Lie triple system. Recently these objects have received attention in the numerical analysis community. A remarkable number of different algorithms can be understood and analyzed using the concepts of symmetric spaces and this theory unifies a range of different topics in numerical analysis, such as polar type matrix decompositions, splitting methods for computation of the matrix exponential, composition of self adjoint numerical integrators and time symmetric dynamical systems. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we are presenting new results related to time reversal symmetries, self adjoint numerical schemes and Yoshida type composition techniques.